Problem $1(20$ points$)$

Consider the following longitudinal model of a viaduct represented in Figure $1.$

$($a$)$ Consider that all elements have constant bending stiffness, EI$.$ Identify the degree of kinematic independency, $,$ and the corresponding independent displacements, in the following situations:

i$.$ all elements have constant axial stiffness, $EA$;

ii$.$ elements CD$,$ DE$,$ EF and FG have constant axial stiffness, $EA,$ and elements AD and FB have axial stiffness, $EA=$;

iii. elements CD$,$ DE$,$ EF and FG have axial stiffness, $EA=,$ and elements AD and FB have constant axial stiffness, $EA.$

Due the symmetry of the geometry, loading, supports and material, the simplification presented in figure $2$ can be used to perform the structural analysis of the structure represented in figure $1.$

$($b$)$ Identify the independent displacements;

$($c$)$ Determine the stiffness matrix, $K,$ and the operator $x_c$ of the complementar solution;

$($d$)$ Determine the fixed$-$end forces vector, $Q_0,$ the nodal forces vector, $Q_N,$ and the operator $x_0$ of the particular solution;

$EI=$ constant $kN{m}^2$

Elements BC and CD

$EA=2EI[kN]$

Element AC

$EA=4EI[kN]$

Structural Mechanics $2$

Continous assessment: Displacement method II

$21$st October $2024$

$($e$)$ Solve the equilibrium equation and verify that displacements at node C are given by

$\{[{}_C],[{}_C^h],[{}_C^v]\}=\frac{1}{EI}\{[189()],[288()],[884(darr)]\}$

$($f$)$ Determine the independent internal forces in all beams;

$($g$)$ Determine the support reactions and represent the internal forces $N,V$ and $M,$ identifying all the necessary parameters to completely define the corresponding functions;

$($h$)$ Qualitatively represent the deformed shape.

Problem $1(20$ points$)$

Consider the following longitudinal model of a viaduct represented in Figure $1.$

$($a$)$ Consider that all elements have constant bending stiffness, EI$.$ Identify the degree of kinematic independency, $,$ and the corresponding independent displacements, in the following situations:

i$.$ all elements have constant axial stiffness, $EA$;

ii$.$ elements CD$,$ DE$,$ EF and FG have constant axial stiffness, $EA,$ and elements AD and FB have axial stiffness, $EA=$;

iii. elements CD$,$ DE$,$ EF and FG have axial stiffness, $EA=,$ and elements AD and FB have constant axial stiffness, $EA.$

Due the symmetry of the geometry, loading, supports and material, the simplification presented in figure $2$ can be used to perform the structural analysis of the structure represented in figure $1.$

$($b$)$ Identify the independent displacements;

$($c$)$ Determine the stiffness matrix, $K,$ and the operator $x_c$ of the complementar solution;

$($d$)$ Determine the fixed$-$end forces vector, $Q_0,$ the nodal forces vector, $Q_N,$ and the operator $x_0$ of the particular solution;

$EI=$ constant $[kN(m{)}^2]$

Elements $BC$ and $CD$

$EA=2EI[kN]$

Element $AC$

$,EA=4EI[kN]$

$($e$)$ Solve the equilibrium equation and verify that displacements at node C are given by

$\{[{}_C],[{}_C^h],[{}_C^v]\}=\frac{1}{EI}\{[189()],[288()],[884(darr)]\}$

$($f$)$ Determine the independent internal forces in all beams;

$($g$)$ Determine the support reactions and represent the internal forces $N,V$ and $M,$ identifying all the necessary parameters to completely define the corresponding functions;

$($h$)$ Qualitatively represent the deformed shape.